How can I remove the lines around the edge of an object?
It appears to be a UV layout problem - not enough margin between the UV islands, so color data is bleeding over. From what I can see of the UV texture on the right everything is just packed way to closely together. But maybe you can fix this by scaling the UV islands in a little, one-by-one. 

Instead of this...



More like this:



Also, to fix triangles like that one on the arm, enable Auto Smooth in the Mesh Context.



Update:

I looked at your .blend and it does appear that the UV layout is the problem.

Note how there is no margin between the islands.



In addition to that, there are parts of the layout that are doubled-up. This may or may not be intentional, but it means a texture in that region will show up on more than one place on the mesh.